theorem Th37:
  f in L1_Functions M & g in L1_Functions M implies (g a.e.= f,M
  iff g in a.e-eq-class(f,M))
proof
  assume
A1: f in L1_Functions M & g in L1_Functions M;
  hereby
    assume g a.e.= f,M;
    then f a.e.= g,M;
    hence g in a.e-eq-class(f,M) by A1;
  end;
  hereby
    assume g in a.e-eq-class(f,M);
    then ex r be PartFunc of X,REAL st g=r & r in L1_Functions M & f in
    L1_Functions M & f a.e.= r,M;
    hence g a.e.= f,M;
  end;
end;
